The 99th Fibonacci Identity
Arthur T. Benjamin, Alex Eustis, Sean S. Plott · The Electronic Journal of Combinatorics · 2008
We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences $G_n$ that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well. Among these, we prove $\sum_{k\geq 0}{n \choose k}G_{2k} = 5^n G_{2n}$ and $\sum_{k\geq 0}{n \choose k}G_{qk}(F_{q-2})^{n-k} = (F_q)^n G_{2n}$.